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Spring Topology and Dyanmical Systems Conference 2010
March 18-20, 2010
Mississippi State University
Starkville, Mississippi, USA

Organizers
Paul Fabel, Wayne Lewis, Frederic Mynard

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On the openness of the induced map Cn(f)
by
Javier Camargo
Universidad Industrial de Santander

A continuum is a nonempty, compact, connected and metric space. A map is assumed to be a continuous function. Given a continuum X, we consider the following hyperspaces of X:


    (1) 2X={A ⊂ X:A is closed and nonempty};
    (2) Cn(X)={A ∈ 2X:A has at most n components}, n ∈ N.
2X is topologized with the Vietoris topology. Furthermore, Cn(X) is a subspace of 2X. Let f:X→ Y be a map between continua. Then the function 2f:2X→ 2Y given by 2f(A)=f(A), for each A ∈ 2X, and Cn(f):Cn(X)→ Cn(Y) defined by Cn(f)=2f|Cn(X) are maps.

In Induced mappings on hyperspaces, H. Hosokawa showed that 2f is open if and only if f is open. Moreover, J. J. Charatonik, A. Illanes and S. Macías, in Induced mappings on the hyperspaces Cn(X) of a continuum X, proved that if Cn(f) is open, then f is also open. Also, they showed that Cn(f) is confluent if and only if f is monotone, for each n ≥ 3. Thus, if Cn(f) is open, where n ≥ 3, then f is open and monotone.

In this talk we show that if the induced map Cn(f) is open, then f is a homeomorphism, for each n ≥ 2.

Date received: February 2, 2010

Copyright © 2010 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cazl-34.